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Closed range theorem

From Wikipedia, the free encyclopedia

In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.

The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Statement

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Let and be Banach spaces, a closed linear operator whose domain is dense in and the transpose of . The theorem asserts that the following conditions are equivalent:

  • the range of is closed in
  • the range of is closed in the dual of

Where and are the null space of and , respectively.

Note that there is always an inclusion , because if and , then . Likewise, there is an inclusion . So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.

Corollaries

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Several corollaries are immediate from the theorem. For instance, a densely defined closed operator as above has if and only if the transpose has a continuous inverse. Similarly, if and only if has a continuous inverse.

Sketch of proof

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Since the graph of T is closed, the proof reduces to the case when is a bounded operator between Banach spaces. Now, factors as . Dually, is

Now, if is closed, then it is Banach and so by the open mapping theorem, is a topological isomorphism. It follows that is an isomorphism and then . (More work is needed for the other implications.)

References

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  • Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
  • Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.